I've managed to show the part, but I'm having trouble with the next part, the . Am I right in thinking that is the number of functions , and is this on the right lines, or am I looking way off?

Any help would be appreciated!

Feb 19th 2010, 09:02 AM

tonio

Quote:

Originally Posted by jackprestonuk

Hi, I've been asked to show the following assertion;

I've managed to show the part, but I'm having trouble with the next part, the . Am I right in thinking that is the number of functions , and is this on the right lines, or am I looking way off?

Any help would be appreciated!

I suppose we're talking of cardinals here, so let us put , and thus by Cantor's Theorem:

, and since is straightforward, applying Cantor-Schroeder-Bernstein we're done.

Tonio

Feb 21st 2010, 11:27 AM

jackprestonuk

Ah, thanks very much, that's really helpful!
In the meantime I'd come up with this argument; I wonder if it works just as well?

Set . Then , and so . We have that and so we have our equality by Cantor Bernstein.

Apologies if sometimes I did or didn't put |...| in there, I still get a little confused when dealing with cardinals whether, say, represents the set of functions from the natural number to itself, or simply the cardinal number (that is, effectively, the number of such functions), or can stand for both...